The mass function f (M) is given by the following equation:
f (M) = MX3sin3i/(Mx + Mc )2 = PK3 /2nG, where Mx and Mc are the masses of the X-ray emitting compact object and of the companion star, respectively, and i is the inclination of the binary orbit. P is the orbital period, and K denotes the amplitude of the Doppler curve that gives the line-of-sight component of the radial velocity of the companion. Both P and K are optically measurable quantities. For HMXBs, the optical light from the companion dominates. For LMXBs, the companions are very faint, hence optical search is practically impossible. When a transient X-ray outburst occurs, it accompanies a nova-like optical brightening, which allows identification of the optical counterpart (see Sect. 16.4.2). After the source returns to X-ray quiescence, the optical light from the companion can be observed.
Once f (M) is obtained, the actual mass of the compact object can be estimated if i and Mc are known. The method of obtaining these quantities from spectroscopic and photometric measurements has been well developed (e.g. ), although sometimes subject to systematic uncertainties.
Cyg X-1 had been the sole black hole candidate for the following 10yrs until two bright X-ray sources in the Large Magellanic Cloud, LMC X-3, and LMC X-1, were added. Both of them were optically identified with early-type stars. The mass functions obtained indicated that the masses of the compact objects in these two sources were also larger than 3M0. These three are all high-mass (Mc >M0) systems.
In 1986, the research of black-hole XBs entered into a new era. McClintock and Remillard  discovered that the compact object of a LMXB transient A0620-00
was definitely more massive than 3 M0. A0620-00 underwent a transient X-ray outburst in 1975. It became the brightest X-ray source in the sky, three times as bright as Sco X-1, and faded back to quiescence (see the lightcurve in Fig. 16.1). The secondary was found to be a K-dwarf (Mc < M0, hence LMXB). Note that, as evident from the equation, f (M) gives an absolute lower limit of the mass of the compact object. Since the measured mass function itself was close to 3 M0 (see Table 16.1), there was very little doubt that Mx > 3M0. This was an epoch-making discovery not only because it presented a convincing case for a black hole, but also because it was the first in low-mass binary systems and furthermore it was a transient source.
In the following 10yrs, X-ray outbursts of several LMXBs were successively detected with the X-ray satellites such as Ginga, Granat, and CGRO. Surprisingly, most of them turned out to contain black holes based on the mass functions (see Table 16.1). In particular, since RXTE became operational in 1996, the detection rate of transients has increased significantly. The RXTE all-sky monitor (ASM) surveys up to —80% of the sky every day, and it is sensitive enough to detect bright X-ray transients practically from the entire Galaxy.
Currently (as of 2005), there are 20 secure black-hole binaries that satisfy Mx > 3M0. They are listed in Table 16.1. Remarkably, there are only three persistent black-hole binaries (Cyg X-1, LMC X-1, and LMC X-3), and they are all HMXBs. The others are all LMXBs, and also transient sources detected during X-ray outbursts (see Table 16.1). As a matter of fact, a large fraction of LMXB transients have turned out to be black-hole binaries (see Sect. 16.4.2). Two black-hole binaries, 4U 1543-47 and SAX J1819.3-2525, have relatively massive (—3Me) secondaries , yet distinctly less massive than O/B-stars.
One finds from Table 16.1 that excluding the three HMXBs, 14 LMXBs out of 17 show f (M) — 3 M0. For them, the f (M) value alone is sufficient to conclude Mx > 3 M0, though additional quantities are required for deriving the actual value of Mx. The evidence for Mx > 3 M0 is also very strong for the remaining three LMXBs. In clear contrast, for those binaries of which the compact objects are known to be neutron stars, the Mx values estimated from f (M) are almost all close to 1.4 M0 (the Chandrasekhar limit, e.g., ).
Was this article helpful?